Collaborative Research: Nonlinear PDE's and Integro-Differential Equations in the Complex Plane

摘要/ Abstract摘要：O Costin，罗格斯大学，非线性偏微分方程与积分微分方程的合作研究。该建议在非线性偏微分方程解的研究中发展和应用新的构造方法，重点研究奇异点的存在唯一性、规律性和自发形成。利用Borel-Laplace正则化和可和性的最新进展，求出偏微分方程解的全局性质。有广泛的问题，这些问题是必不可少的，一些具体的例子，包括薄膜方程，Kuramoto-Sivashinski， KdV和mKdV将被分析。物理、化学、材料科学和许多其他学科中的自然现象最常用微分方程来描述；复杂现象常常用偏微分方程来模拟。本项目涵盖了广泛的此类方程，其解的整体性质及其可能的爆破。这项研究是一项正在展开的事业的一部分，以推进奇异行为理论，在数学及其应用中具有重要意义。

Abstract: 0406193 O Costin, Rutgers UniversityCollaborative research on nonlinear PDEs and integro-differential equations The proposal develops and applies new constructive methods in the study of solutions of nonlinear partial differential equations, with focus on existence and uniqueness, regularity and spontaneous formation of singularities. Recent advances in Borel-Laplace regularization andsummability are used to find global properties of solutions ofPDEs. There is a wide range of problems where these questions areessential, and a number of concrete examples including thin filmequations, Kuramoto-Sivashinski, KdV and mKdV will be analyzed.Natural phenomena in physics, chemistry, material sciences and in manyother disciplines are most often described by differential equations;complex phenomena are frequently modeled by partial differentialequations. The present project covers a wide range of such equations,the global properties of their solutions and their possible blowup. This resarch is a piece of an unfolding enterprise to advance atheory of singular behavior, of substantial importance in mathematicsand its applications.